First, galaxy overdensity is related to matter density contrast through:
[tex]\delta_{\rm{g}}=b*\delta_{\rm{m}}[/tex], where [tex]b[/tex] is the bias. [tex]\delta_{\rm{m}}[/tex] cannot be less than -1, because it is defined as [tex](\rho-\overline{\rho})/\overline{\rho}[/tex], since [tex]\rho \geq 0[/tex], [tex]\delta_{\rm{m}} \geq -1[/tex].

However, [tex]b[/tex] can take any value. It can be [tex]-3, -2[/tex] (void), or very positive number, such as [tex]2[/tex] in Table 5 and 6 of 1303.4486. Therefore, [tex]\delta_{\rm{g}}[/tex] can be any value. For example, If [tex]\delta_{\rm{m}}=-0.8[/tex], [tex]b=2[/tex], then [tex]\delta_{\rm{g}}=-1.6[/tex]. Then what is the definition of [tex]\delta_{\rm{g}}[/tex] if it can take any value?

It certainly cannot be defined as the matter density contrast because it can be less than -1. Then how to understand its physical meaning?

A negative bias would be totally unphysical when one concerns density. However an anti-bias (0<b<1) is possible and in the fact is present for small galaxies/haloes.

As Maciej had noticed You have used a definition of the linear bias. In general we have:

[tex]\delta_g= f(\delta)[/tex]

in particular we can express the non-linear function f in Tylor series:
[tex]\delta_g = \sum_{k=0}^\infty {b_k\over k!} \delta^k[/tex]
(see e.g. Fry&Gaztanaga 1994)
hence in regions where [tex]b_1>1[/tex] and [tex]\delta<0.8[/tex] rest of the bias parameters will have values making the [tex]\delta_g\geq -1[/tex] in the end.

As Maciej and Wojciech said, the discussion concerns linear perturbation theory, meaning [tex]|\delta| \ll 1[/tex]. Your value of [tex]\delta_m=-0.5[/tex] is highly non-linear, so the linear theory is no longer valid. If you apply it anyway, then you get unphysical results.

Similarly, when the overall virialisation fraction at low redshifts [tex]f_{vir}(z)[/tex] fails to satisfy [tex]f_{vir}(z) \ll 1[/tex], the underlying homogeneity assumption fails and an artefact - would-be "dark energy" - arises if the homogeneous (FLRW) metric is used to interpret the observations despite the invalidity of the homogeneity assumption (1303.4444).